Let f(x) = 2(x- h1)*2 + k1 and g(x) = ( x- h2)*2 + k2, where h1, h2,k1,k2. The vertex of the graph of f is at (m,-m*2) and the vertex of the graph of g is at (-m,-m), where 0 < m < 1. The graphs of f and g intersect at exactly one point. Find the value of m.
To find the value of m, we need to first find the equations of f(x) and g(x) using the information given.
From the vertex of f, we know that (h1, k1) = (m, -m*2), so the equation of f(x) can be written as:
f(x) = 2(x - m)^2 - 2m
From the vertex of g, we know that (h2, k2) = (-m, -m), so the equation of g(x) can be written as:
g(x) = (x + m)^2 - m
We are given that the graphs of f and g intersect at exactly one point, so we can set f(x) = g(x) and solve for x:
2(x - m)^2 - 2m = (x + m)^2 - m
Expanding both sides:
2(x^2 - 2mx + m^2) - 2m = x^2 + 2mx + m^2 - m
2x^2 - 4mx + 2m^2 - 2m = x^2 + 2mx + m^2 - m
Simplifying and rearranging:
x^2 - 6mx + m^2 + m - 2m^2 + 2m = 0
x^2 - 6mx - m^2 = 0
Since we are given that the graphs intersect at exactly one point, the discriminant of this quadratic equation must be equal to zero:
b^2 - 4ac = (-6m)^2 - 4(1)(-m^2) = 0
36m^2 + 4m^2 = 0
40m^2 = 0
m^2 = 0
m = 0
Since the given condition is that 0 < m < 1, the value of m that satisfies all the conditions is m = 0.